A dynamic approach to the tendency of industries to cluster



From (6) we see that:


(9)


( nι

Z =

V i=1


1-1   ъ ■ - A σ-1

x1iσ +x2Γ

i =1        )


Making a substitution of (7) and (8) into (9) leaves us with:


(10)


Z = xrnσ

1 11

j V Pj 1 )


Collecting the terms in (10) that relate to input xi 11 and simplifying leads to:


(11)


Z = xi11 Pn P j1σ
V j


(_

+ xi11 Pil Σ (Pj2 t)
V j


1-σ

)


σ

Aσ-1


This can be written as:
(12)
Z = xn1 Ph P - -

1

( n1             n2               A 1-n

, where p = I p}1 σ +(pj21)1-σ I

Due to the symmetric fashion in which the


V i =1            i =1                )

producer service varieties enter the sub-production function of the manufacturing firm,
this expression of
P can be simplified to read:
(13)
p = (n 1 p 1-σ + n2( p21)1-σ )1-σ

, which the same as Equation (2). To see that P is the cost function, we solve for input
demand:
(14)
xfl1 = ZPi-σ P p

Letting the cost function be evaluated at a production of Z=1 leads to:

-σ

(15) c = P11-σPσ +(Pi21)1-σ Pσ = Pσ (n 1 p1-σ + n2(P21)1-σ )1-σ = PσP1-σ = P

i                                         i

23



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